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Strong uniform continuity. (English) Zbl 1161.54003
Strong local continuity is a relative concept: if \(f:X\to Y\) is a continuous map between metric spaces then not only is \(f\upharpoonright K\) uniformly continuous whenever \(K\) is compact: the \(\delta>0\) that corresponds to the given \(\epsilon>0\) satisfies the implication “if \(d(x,y)<\delta\) then \(d(f(x),f(y))<\epsilon\)” even when just one of \(x\) and \(y\) belongs to \(K\). This state of affairs is abbreviated as: \(f\) is strongly uniformly continuous on \(K\). The authors study this concept in some depth. They compare the families \(\mathcal{B}^f=\{B:f\upharpoonright B\) is uniformly continuous\(\}\) and \(\mathcal{B}_f=\{B:f\) is strongly uniformly continuous on \(B\}\); the latter is an ideal (and a bornology if \(f\) is continuous), the former need not be.
In the second part of the paper the attention shifts to function space topologies; for a bornology \(\mathcal{B}\) the authors study the topology of strong uniform convergence on members of \(\mathcal{B}\) (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of \(\mathcal{B}\)).
Reviewer: K. P. Hart (Delft)

MSC:
54C05 Continuous maps
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C35 Function spaces in general topology
54E15 Uniform structures and generalizations
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[1] Arens, R., A topology for spaces of transformations, Ann. of math., 47, 480-495, (1946) · Zbl 0060.39704
[2] Arens, R.; Dugundji, J., Topologies for function spaces, Pacific J. math., 1, 5-31, (1951) · Zbl 0044.11801
[3] Atsuji, M., Uniform continuity of continuous functions of metric spaces, Pacific J. math., 8, 11-16, (1958) · Zbl 0082.16207
[4] Attouch, H.; Lucchetti, R.; Wets, R., The topology of the ρ-Hausdorff distance, Ann. mat. pura appl., 160, 303-320, (1991) · Zbl 0769.54009
[5] Beer, G., More about metric spaces on which continuous functions are uniformly continuous, Bull. austral. math. soc., 33, 397-406, (1986) · Zbl 0573.54026
[6] Beer, G., Conjugate convex functions and the epi-distance topology, Proc. amer. math. soc., 108, 117-126, (1990) · Zbl 0681.46014
[7] Beer, G., Topologies on closed and closed convex sets, (1993), Kluwer Acad. Publ. Dordrecht · Zbl 0792.54008
[8] Beer, G., On metric boundedness structures, Set-valued anal., 7, 195-208, (1999) · Zbl 0951.54025
[9] Beer, G.; Di Concilio, A., Uniform continuity on bounded sets and the attouch – wets topology, Proc. amer. math. soc., 112, 235-243, (1991) · Zbl 0677.54007
[10] G. Beer, S. Levi, Total boundedness and bornologies, Topology Appl., in press · Zbl 1167.54008
[11] Beer, G.; Levi, S., Pseudometrizable bornological convergence is attouch – wets convergence, J. convex anal., 15, 439-453, (2008) · Zbl 1173.54002
[12] Borsik, J., Mappings that preserve Cauchy sequences, Časopis pro Pěstování mat., 113, 280-285, (1988) · Zbl 0657.54024
[13] Brandi, P.; Ceppitelli, R., A new graph topology. connections with the compact-open topology, Appl. anal., 53, 185-196, (1994) · Zbl 0836.54010
[14] P. Brandi, R. Ceppitelli, L. Hola, Boundedly UC spaces, hypertopologies and well-posedness, Rap. tecnico N. 13, Dip. Math. e Inf., University of Perugia, 2001
[15] Caterino, A.; Panduri, T.; Vipera, M., Boundedness, one-point extensions, and B-extensions, Math. slovaca, 58, 1, 101-114, (2008) · Zbl 1164.54018
[16] Di Concilio, A.; Naimpally, S., Proximal convergence, Monatsh. math., 103, 93-102, (1987) · Zbl 0607.54013
[17] Di Maio, G.; Meccariello, E.; Naimpally, S., Decompositions of UC spaces, Questions answers gen. topology, 22, 13-22, (2004) · Zbl 1060.54016
[18] Di Maio, G.; Naimpally, S., Proximal graph topologies, Questions answers gen. topology, 10, 97-125, (1992) · Zbl 0802.54009
[19] Dugundji, J., Topology, (1989), Wm.C. Brown Publishers Dubuque, IA
[20] Engleking, R., General topology, (1977), Polish Scientific Publishers Warsaw
[21] Hogbe-Nlend, H., Bornologies and functional analysis, (1977), North-Holland Amsterdam · Zbl 0359.46004
[22] Hu, S.-T., Boundedness in a topological space, J. math. pures appl., 228, 287-320, (1949) · Zbl 0041.31602
[23] Klein, E.; Thompson, A., Theory of correspondences, (1984), Wiley New York
[24] Jain, T.; Kundu, S., Atsuji completions: equivalent characterisations, Topology appl., 154, 28-38, (2007) · Zbl 1110.54015
[25] Lechicki, A.; Levi, S.; Spakowski, A., Bornological convergences, J. math. anal. appl., 297, 751-770, (2004) · Zbl 1062.54012
[26] Levine, N.; Saunders, W., Uniformly continuous sets in metric spaces, Amer. math. monthly, 67, 153-156, (1960) · Zbl 0087.37703
[27] Lucchetti, R., Convexity and well-posed problems, (2006), Springer-Verlag New York · Zbl 1106.49001
[28] McCoy, R.; Ntantu, I., Topological properties of spaces of continuous functions, (1988), Springer-Verlag Berlin · Zbl 0647.54001
[29] Naimpally, S.; Warrack, B., Proximity spaces, (1970), Cambridge Univ. Press Cambridge · Zbl 0206.24601
[30] Snipes, R.F., Functions that preserve Cauchy sequences, Nieuw arch. voor wiskd., 25, 409-422, (1977) · Zbl 0376.54011
[31] Willard, S., General topology, (1970), Addison-Wesley Reading, MA · Zbl 0205.26601
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